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The diagram below consists of a small square, four equilateral triangles, and a large square.  Find the area of the large square.

 

 May 4, 2024
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Let's denote the side length of the small square as "s".

Equilateral Triangles:

Since the triangles are equilateral, each side length is equal to "s". The height of an equilateral triangle with side length "s" is given by 23​s​.

Large Square:

The large square consists of the small square in the center and four right triangles formed by the sides of the equilateral triangles and the sides of the large square.

Finding the Side Length of the Large Square:

The base of each right triangle is "s" (side of the small square).

The height of each right triangle is the height of the equilateral triangle, which is 23​s​.

By the Pythagorean Theorem, the hypotenuse (which is the side of the large square) can be found as:

Side of Large Square^2 = s^2 + (sqrt(3)s/2)^2 Side of Large Square^2 = s^2 + (3s^2)/4 Side of Large Square^2 = (7s^2)/4 Side of Large Square = sqrt((7s^2)/4) Side of Large Square = (s * sqrt(7))/2

Area of the Large Square:

The area of a square is calculated by squaring the side length. Therefore, the area of the large square is:

Area of Large Square = (Side of Large Square)^2 Area of Large Square = ((s * sqrt(7))/2)^2 Area of Large Square = (s^2 * 7) / 4

Therefore, the area of the large square is a function of the small square's side length "s" and is proportional to 7/4 times the area of the small square.

 May 4, 2024

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